Method for determining a heat source model for a weld

ABSTRACT

A method for determining a heat source model for a weld. The method includes the steps of determining a double elliptical distribution of the heat density of the weld, modifying the double elliptical distribution as a function of a profile geometry of the weld, and determining the heat source model as a function of the modified double elliptical distribution.

This application claims the benefit of prior provisional patentapplication Serial No. 60/157,244 filed Oct. 1, 1999.

TECHNICAL FIELD

This invention relates generally to a method for determining a heatsource model for a weld and, more particularly, to a method fordetermining a heat source model for an arbitrarily shaped weld profile.

BACKGROUND ART

It is imperative when welding a joint between two metals, for example,in gas metal arc welding, to minimize distortions and residual stresses,and to maximize the strength of the welded joint and the surroundingstructure.

One method for determining the residual stress of a weld joint is tomodel the stress based on a modeled weld heat source. For example, acommon and very popular method for modeling a weld heat source is knownas a double elliptical heat source model, disclosed by Goldak et al.(Goldak) in a paper entitled, A New Finite Element Model for WeldingHeat Sources (Metallurgical Transactions, Volume 15B, June, 1984, pages299-305).

The model disclosed by Goldak, however, may not be directly applied toan arbitrarily shaped weld profile. For example, welding procedures,parameters of the materials, and joint types, e.g., T-fillet joints,butt joints, lap joints, and the like, result in weld heat sources thatare no longer elliptical. Therefore, it would be desired to incorporatethe weld profile geometry of the joint into the heat source model toimprove the accuracy of the model.

The present invention is directed to overcoming one or more of theproblems as set forth above.

DISCLOSURE OF THE INVENTION

In one aspect of the present invention a method for determining a heatsource model for a weld is determined. The method includes the steps ofdetermining a double elliptical distribution of the heat density of theweld, modifying the double elliptical distribution as a function of aprofile geometry of the weld, and determining the heat source model as afunction of the modified double elliptical distribution.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagrammatic illustration of a butt joint;

FIG. 2 is a diagrammatic illustration of a T-fillet joint;

FIG. 3 is a diagrammatic illustration of a lap joint;

FIG. 4 is a diagrammatic illustration of a modified double ellipsoidheat source model of the T-fillet joint of FIG. 2; and

FIG. 5 is a diagrammatic illustration of a cross section of the weldportion of the T-fillet joint of FIG. 4.

BEST MODE FOR CARRYING OUT THE INVENTION

The present invention provides a method for determining an enhanced heatsource model for a weld by incorporating the weld profile geometry inthe mathematical formulation for a double elliptical heat source model.The invention is described with reference to the accompanying drawingsand illustrations.

The double elliptical heat source model is an advanced weld heat sourcemodel. The double elliptical heat source model assumes that the heatinput from the weld arc is distributed in a volume with doubleellipsoidal geometry. The major advantage of the double elliptical heatsource model over other surface or spherical models is that it offersmore flexibility in modeling weld penetration due to arc digging andstirring.

However, typical weld cross section profiles are not elliptical inshape. For example, with reference to FIGS. 1-3, exemplary weld profilesare shown for some commonly used welds. FIG. 1 illustrates a half-filledbutt joint 100, in which a first piece 102 is attached to a second piece104 by means of a butt joint weld 106. FIG. 2 displays a T-fillet joint200, in which a first piece 202 is attached to a second piece 204 bymeans of a T-fillet weld 206. In like manner, FIG. 3 illustrates a lapjoint 300, in which a first piece 302 is attached to a second piece 304by means of a lap joint weld 306. It is to be understood that the weldjoints of FIGS. 1-3 are merely examples of weld joints that may be usedwith the present invention. Virtually any type of weld joint used maybenefit from use of the present invention.

Referring now to FIGS. 4 and 5, the T-fillet joint 200 of FIG. 2 isshown in greater detail.

The heat source model assumes the following Gauss distribution for theheat density distribution in the weld pool: $\begin{matrix}{{q\left( {\xi,r,\theta} \right)} = {q_{m\quad a\quad x}^{- {k{(\frac{\xi}{c})}}^{2}}^{- {k{(\frac{r}{r_{b}{(\theta)}})}}^{2}}}} & \left( {{Equation}\quad 1} \right)\end{matrix}$

where q_(max) is the maximum heat density at an arc center 404 of theT-fillet joint weld; ξ is a moving coordinate in the welding direction,i.e., ξ=z−νt+z₀; v is an arc traveling speed; z₀ is the distance fromthe arc starting position to the cross sectional plane of interest for atwo dimensional cross sectional model, or to the plane z=0 for a threedimensional model; r_(b)(θ) is the distance from the heat center topoints 1, 2, . . . k, k+1, . . . on the boundary of the weld profile; rand θ are local polar coordinates at the weld cross section; c is thesemi-axis of the weld pool in the welding direction; and k is aparameter known as a heat concentration coefficient used to determineheat density at the boundary of weld cross section, i.e.,${{k = {\ln \left( \frac{q_{m\quad a\quad x}}{q_{b}} \right)}}}_{\xi = 0}.$

It is assumed, with reference to FIG. 4, that the heat source model 402forms double elliptical distributions for the heat density in thewelding, i.e., longitudinal, direction with front semi-axis c₁ and rearsemi-axis c₂. Therefore, $\begin{matrix}{{q_{1}\left( {\xi,r,\theta} \right)} = {{q_{m\quad a\quad x}^{- {k{(\frac{\xi}{c_{1}})}}^{2}}^{- {k{(\frac{r}{r_{b}{(\theta)}})}}^{2}}\quad {for}\quad \xi} \geq 0}} & \left( {{Equation}\quad 2} \right) \\{and} & \quad \\{{q_{2}\left( {\xi,r,\theta} \right)} = {{q_{m\quad a\quad x}^{- {k{(\frac{\xi}{c_{2}})}}^{2}}^{- {k{(\frac{r}{r_{b}{(\theta)}})}}^{2}}\quad {for}\quad \xi} < 0.}} & \left( {{Equation}\quad 3} \right)\end{matrix}$

The continuity condition for the heat density at ξ=0

q ₁(0,rθ)=q ₂(0, r,θ)  (Equation 4)

holds from Equations 2 and 3.

The total heat input in the model, preferably in Joules per second, canbe related to the welding parameters as follows: $\begin{matrix}\begin{matrix}{Q = {{\eta \quad {VI}} = \quad {{\int_{0}^{\infty}{\int_{0}^{\theta_{n}}{\int_{0}^{\infty}{{q_{1}\left( {\xi,r,\theta} \right)}\quad r{\theta}{r}{\xi}}}}} +}}} \\{\quad {\int_{0}^{\infty}{\int_{0}^{\theta_{n}}{\int_{0}^{\infty}{{q_{2}\left( {\xi,r,\theta} \right)}r{\theta}{r}{\xi}}}}}} \\{= \quad {q_{m\quad a\quad x}\frac{\sqrt{\pi}\left( {c_{1} + c_{2}} \right)}{4k\sqrt{k}}{\int_{0}^{\theta_{n}}{{r_{b}^{2}(\theta)}{\theta}}}}} \\{= \quad {q_{m\quad a\quad x}\frac{\sqrt{\pi}\left( {c_{1} + c_{2}} \right)}{4k\sqrt{k}}\Psi}}\end{matrix} & \left( {{Equation}\quad 5} \right)\end{matrix}$

where Ψ is a function of weld profile geometry only. Preferably, theweld profile can be represented by a discrete number of points (1, 2, .. . k, k+1, . . . ) as shown in FIG. 5. The function Ψ, in the preferredembodiment, is expressed as $\begin{matrix}\begin{matrix}{\Psi = \quad {\sum\limits_{k = 2}^{n}\quad {\int_{\theta_{k - 1}}^{\theta_{k}}{{r_{b}^{2}(\theta)}{\theta}}}}} \\{= \quad {{\sum\limits_{k = 2}^{n}\quad \left( {2A_{k}} \right)} = {2A_{w}}}}\end{matrix} & \left( {{Equation}\quad 6} \right)\end{matrix}$

where A_(w) is the area of the fusion region. From Equations 5 and 6, itmay be determined that $\begin{matrix}{q_{m\quad a\quad x} = {\frac{2k\sqrt{k}}{\sqrt{\pi \quad}{A_{w}\left( {c_{1} + c_{2}} \right)}}\eta \quad {{VI}.}}} & \left( {{Equation}\quad 7} \right)\end{matrix}$

In the special case of an elliptical weld cross section, A_(w)=πab/2,and therefore $\begin{matrix}{q_{m\quad a\quad x} = {\frac{4k\sqrt{k}}{\pi \sqrt{\pi \quad}{{ab}\left( {c_{1} + c_{2}} \right)}}\eta \quad {{VI}.}}} & \left( {{Equation}\quad 8} \right)\end{matrix}$

In the thermal analysis, the initial nodal temperature in the weld areais often assigned to be the melting temperature. Thus, the heat densitydistribution becomes $\begin{matrix}{{q\left( {\xi,r,\theta} \right)} = {\frac{2k\sqrt{k}\left( {{\eta \quad {VI}} - Q_{w}} \right)}{\sqrt{\pi \quad}{A_{w}\left( {c_{1} + c_{2}} \right)}}^{{- {k{(\frac{\xi}{c})}}^{2}}^{- {k{(\frac{r}{r_{b}{(\theta)}})}}^{2}}}}} & \left( {{Equation}\quad 9} \right) \\{where} & \quad \\{Q_{w} = {\rho \quad A_{w}v{\int_{T_{o}}^{T_{m}}{\overset{\_}{c}\quad {T}}}}} & \left( {{Equation}\quad 10} \right)\end{matrix}$

and {overscore (c)} is the temperature dependent specific heat of theweld material.

INDUSTRIAL APPLICABILITY

The present invention is an enhanced welding heat source model that canbe used for an arbitrarily-shaped fusion profile. The model is based ona modification of a double elliptical heat source model with an explicitrepresentation of the weld profile geometry in the mathematicalformulation. The shape of weld profile is discretized by a set of pointsthat can be coincided with nodes in a finite element model. The Gaussiandistribution function is modified in such a way that the heat density ata point inside the weld is proportional to the radial distance from thearc center to the weld profile point. Based on these assumptions,mathematical formulations are derived. The heat source model isdetermined from these mathematical formulations, and is used todetermine distortions, residual stresses, and potential reductions instrength of the weld.

Other aspects, objects, and features of the present invention can beobtained from a study of the drawings, the disclosure, and the appendedclaims.

What is claimed is:
 1. A method for determining a heat source model for a weld, including the steps of: determining a double elliptical distribution of the heat density of the weld; modifying the double elliptical distribution as a function of a profile geometry of the weld; and determining the heat source model as a function of the modified double elliptical distribution of the heat density.
 2. A method, as set forth in claim 1, wherein the profile geometry of the weld is a function of the weld joint.
 3. A method, as set forth in claim 2, wherein the weld joint is a butt joint.
 4. A method, as set forth in claim 2, wherein the weld joint is a lap joint.
 5. A method, as set forth in claim 2, wherein the weld joint is a T-fillet joint.
 6. A method, as set forth in claim 1, wherein the double elliptical distribution of the heat density is a Gauss distribution.
 7. A method, as set forth in claim 2, wherein the profile geometry of the weld is determined as a discrete number of points along the weld joint.
 8. A method, as set forth in claim 7, including the step of correlating the discrete number of points with a set of nodes in a finite element model.
 9. A method, as set forth in claim 1, further including the step of determining residual stresses and distortions of the weld as a function of the heat source model. 